Theorem sbcbi2 3056

Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Assertion

Ref	Expression
sbcbi2	|- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Proof of Theorem sbcbi2

Step	Hyp	Ref	Expression
1		abbi 2321	. . 3 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2		eleq2 2271	. . 3 |- ( { x | ph } = { x | ps } -> ( A e. { x | ph } <-> A e. { x | ps } ) )
3	1, 2	sylbi 121	. 2 |- ( A. x ( ph <-> ps ) -> ( A e. { x | ph } <-> A e. { x | ps } ) )
4		df-sbc 3006	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
5		df-sbc 3006	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6	3, 4, 5	3bitr4g 223	1 |- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   [.wsbc 3005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-sbc 3006

This theorem is referenced by:  csbeq2  3125